Showing the Relation between Counting and Adding:1. Draw 2 small circles on the board
◦ ◦
2. Count them, pointing as you count: "1...2..."3. Let the child count them. "Your turn."4. Define adding so that it ties in with what the child knows about counting. "If I add 1 more, I'll have...:" Draw another circle, and count them.◦ ◦ ◦
"Three circles." Repeat. "I have 3 circles. If I add one more, I'll have..."◦ ◦ ◦ ◦"...1...2...3...4. I'll have 4 circles. If I add 1 more, I'll have..."◦ ◦ ◦ ◦ ◦And so on. Add circles until you reach 20. Repeat the procedure several times.As soon as the child catches on to the idea that when you say, "Add 1,:" you're going to the next number in the counting series, start speeding up the operation. "I have 7 circles and if I have one more, I'll have...yes, 8. I have 8 circles. If I add 1 more I'll have...Good."5. Present the rule for adding 1: Draw 5 circles on the board. "Here's the rule for adding 1. You first see how many you start out with. How many are here? Sure, 5. And you say, "I have 5. If I add 1 more I'll have...if you don't know ask yourself, what comes after 5? What's the next number? Sure, 6. So if I have 5 and add 1 more, I'll have 6."
Reading addition problems:
1. Present 3 + 2 = 5.2. Identify the + as plus, the = as the equal sign.3. Then read, "We read just the way we do in the book. We start over here [left] and say, 3 ... plus ... 2 ... equals ... 5—3 plus 2 equals 5. Your turn. ..."4. Point out that you can add any two numbers together. Demonstrate with various examples. Let the child give the numbers and you write them down. "Pick any numbers you want."5. After he can read addition problems and properly identify the terms (which may take at least an entire session), show him how to interpret addition problems. Begin by pointing out that in addition you start out with on number and end up with another. Put the problem 3 + 2 = 5 on the board. "The problem tells us that we start out with 3 and end up with 5." Give the child practice on several other problems. Then present the problem
4 + 5 ="What about this one? We start out with 4 and we end up with—you don't know. That's what we have to figure out. I know that when you start out with 4 and add 5 you end up with 9, so I can write . . ."4 + 5 = 9"I solved the problem. Read it . . ."Repeat with examples until the child catches on to the idea of starting out and ending up. This is an extremely important notion. Make sure he learns it.
Translate symbols into the operation of adding.
3 + 1 =
"Look at what the problem tells you. It tells you to start out with 3. Then the plus sign tells you to add. It says, Get more numbers. How many more? [Point to the 1.] Look. It says get 1 more. Start out with 3 and get 1 more. And when you do that, you'll end up with 4.
3 + 1 = 4
"Read it."
Keep the presentation straight and simple: start out with the number on the left (at least in the present problems); the plus sign tells you to get more, the number after the plus sign tells you how much more; and the number after the equal sign tells you what you end up with.
6. Diagram the problem.
◦ ◦ ◦ + ◦ =◦ ◦ ◦ ◦
The top row shows you what you start out with and what you add. The bottom row shows what you end up with. The bottom row is equal to the top row because you can draw a line from every circle in the top row to a corresponding circle in the bottom row.
◦ ◦ ◦ + ◦ =| | | |◦ ◦ ◦ ◦
To find out how many you end up with, simply count the symbols in the bottom row. You can see where each one came from. You can see the relations between the three terms.
Stress the idea that the term you start out with and the term you add are reflected in the answer. Point to the bottom row, "See, here's the 3. And here's the 1. 3 plus 1 is 4."
Rule: For every circle in the top row there must be a circle in the bottom row.
7. After the child understands how to read an addition problem and how to interpret it (which will probably take three sessions and 40-50 examples), introduce the rule for adding 1 to other numbers.
3 + 1 =
"What do we start out with? . . . Yes, 3. And what do we do (pointing to the + sign)? . . . Yes, get more. And how many do we add? . . . Yes, 1. . . . And what do we end up with? . . . We don't know. But I know a way to figure it out. Is there a 1 in this problem? . . . See if you can find it. . . . Yes. So I draw a circle around the 1.
3 + 1⃝ =
"We're adding 1 to which number? . . . To 3. [Pointing to the 3.] When you add 1 to a number, you end up with the next number. We're adding 1 to 3 so we're going to end up with the number that comes after 3. What comes after 3? . . . Sure, 4. So we're going to end up with 4. Let's write it down."
3 + 1⃝ = 4
Summarize the problem as an addition fact. "3 plus 1 is 4. Say it."
8. Follow with a series of problems. Include some in which the 1 is first (1 + 3 =) and last (7 + 1 =). Have the child read the problem. "7 plus 1 is . . . we don't know." Then ask, "Are we adding 1?" Have him locate it. Circle it. "What's the rule when you add 1?" Help him state the rule. "When you add 1 to a number, you always end up with the next number." Have him find the number to which 1 is being added. He should have no trouble, because it's the only uncircled number in the problem."What comes after 7? Good. So we'll end up with 8." Write the answer.
7 + 1⃝ = 8
Diagram the problem with circles.
9. Don't introduce 1 + 1 or 1 + 0 until after the child has worked on +1 problems for a week or so. Solve them the same way you would any other +1 problem. In 1+1=, circle only a single 1 and ask what comes after the other 1. In 1+0 =, explain how to handle the zero )which the child already knows vaguely from counting backward). "We call this zero. And zero means that you have nothing. When I say I have zero shoes, it means I don't have any shoes." Give other examples. Then tell the child that 1 comes after 0. Tie this in with what he knows about counting backwards.
10. Demonstrate that the rule about the next number applies only when you're adding 1. The child will probably get the idea after the first few sessions that addition is simpler than it actually is. Find the 1; look at the other number; then say the number that comes next. Pretty easy. To keep him honest, present a series like this.
1 + 4 =
5 + 4 =5 + 1 =6 + 11 =1 + 9 =
"Now remember, we can't use the rule for adding 1 unless we find a 1 in the problem. Do we find a 1 ion the first problem? Good. Then we can use the rule. Let's do it. Do we find a rule in the second problem? No. So we can't use the rule. The problem 5 + 4 is 9 is one you just sort of have to remember. Say it . . ."
11. Present continuity exercises. For the next month, spend about five minutes at the beginning of each session on the +1 problems. Here are several series of problems you can present. They show the relation between +1 and counting.
0 + 1 = 9 + 1 =
1 + 1 = 8 + 1 =2 + 1 = 7 + 1 =3 + 1 = 6 + 1 =4 + 1 = 5 + 1 =5 + 1 = 4 + 1 =6 + 1 = 3 + 1 =7 + 1 = 2 + 1 =8 + 1 = 1 + 1 =9 + 1 = 0 + 1 =
Stress the addition facts "2 plus 1 is . . . well, what comes after 2? . . . 3. So, 2 plus 1 is 3. Say it . . . 3 plus 1 is . . . What comes after 3? . . . 4. So 3 plus 1 is 4. Say it . . ." These are quite similar to the What comes next? games the child played during the previous year.
Word Problems Involving Adding 1: Introduce word problems after the child has been exposed to +1 problems for at least two weeks. The idea to get across is that the rule for adding works with dogs, sheep, cows, or anything else the child can name. The idea is not that these prove the rule. You wouldn't try to convince the child that a red ball proves the concept red. It is merely an example of red, just as adding pies is an example of adding.
Present problems like these: "I have 8 dollars. Then I get 1 more. How many do I have?" "A farmer buys 4 sheep. Then he buys 1 more. How many does he have?"
1. Tell the child specifically how to translate the words into mathematical symbols. The parallel between the every-day language we use and the language of mathematics is not obvious—not even logical. Conventions: The first number you mention is the one you start out with. If the first number mentioned is 7, you start out with 7. The words more, adds, gets, buys tells you that you're adding. You indicate these on the chalkboard with a plus sign. Then you add as many as the farmer gets or adds or buys. If he steals 1, you write 7 + 1. Next, you put an equal sign, and ask the question the problem poses.
7 + 1 =
How many does he end up with? We don't know. That's what we're going to figure out.
2. Diagram the problem on the board, using circles for each number in the problem. If the farmer has 8 bulls and buys 1 more, diagram the problem this way:
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ + ◦ =
| | | | | | | | |◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
Explain that each circle stands for a bull. "For every bull we have in the top row we must have 1 in the bottom, so we draw them in the bottom row and count them—that's how many bulls the farmer has."
Introduce one or two word problems during each session for at least two weeks. Don't be afraid of repeating problems framed in the same situation (such as bull-buying). The only way the child will learn the key words is to hear them over and over.
That is about 6 pages, ending at p. 202, of a 300 page paperback. The instructions are much more dense than "Teach your child to read," but I was able to follow them without confusing myself or confusing her. It took two or three weeks to get through it, half an hour a day at the dining table with a tiny whiteboard.
The rest of page 202 is another outline, even more compressed, to teach the number pairs: 1 + 1 through 5 + 5 at first, and 6 + 6 through 10 + 10 after another two weeks. On page 203 a new section starts with the heading "Algebra Problems."
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